### Lagrange-type Algebraic Minimal Bivariate Fractal Interpolation Formula

Received: 28 July 2023    Accepted: 14 September 2023    Published: 25 September 2023
Abstract

Fractal interpolation methods became an important method in data processing, even for functions with abrupt changes. In the last few decades it has attracted several authors because it can be applied in various fields. The advantage of these methods are that we can generalize the classical approximation methods and also we can combine these methods for example with Lagrange interpolation, Hermite interpolation or spline interpolation. The classical Lagrange interpolation problem give the construction of a suitable approximate function based on the values of the function on given points. These method was generalized for more than one variable functions. In this article we generalize the so-called algebraic maximal Lagrange interpolation formula in order to approximate functions on a rectangular domain with fractal functions. The construction of the fractal function is made with a so-called iterated function system. This method it has the advantage that all classical methods can be obtained as a particular case of a fractal function. We also use the construction for a polynomial type fractal function and we proof that the Lagrange-type algebraic minimal bivariate fractal function satisfies the required interpolation conditions. Also we give a delimitation of the error, using the result regarding the error of a polynomial fractal interpolation function.

 Published in Applied and Computational Mathematics (Volume 12, Issue 5) DOI 10.11648/j.acm.20231205.11 Page(s) 109-113 Creative Commons This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. Copyright Copyright © The Author(s), 2023. Published by Science Publishing Group
Keywords

Fractal Interpolation, Lagrange Interpolation, Fractal Surfaces

References
 [1] M. F. Barnsley, Fractal Function and Interpolation, Constr. Approx., 1986, vol. 2, no. 4, pp. 303-329. [2] M. F. Barnsley, A. N. Harrington, The Calculus of Fractal Interpolation Functions, J. Approx. Theory, 1989, vol. 57, no. 1, pp 14-34. [3] M. F. Barnsley, M. Hegeland and P. Massopust, Numerics and Fractals, https://arxiv.org/abs/1309.0972, 2014. [4] A. K. B. Chand, G. P. Kapoor, Generalized Cubic Spline Fractal Interpolation Functions, SIAM Journal on Numerical Analysis, Vol. 44, No. 2, 2006, pp. 655-676. [5] A. K. B. Chand, M. A. Navascués, Generalized Hermite Fractal Interpolation, Academia de Ciencias, Zaragoza, Vol. 64, 2009, pp. 107-120. [6] A. K. B. Chand, M. A. Navascués, P. Viswanathan, S. K. Katiyar, Fractal trigonometric polynomial for restricted range approximation Fractals, 24(2), 2016, 1-11. [7] Gh. Coman, M. Birou, T. Catinas, C. Osan, A. Oprisan, I. Pop, I. Somogyi, I. Todea, Interpolation Operators, Casa Cartii de Stiinte, Cluj, 2004. [8] L. Dalla, Bivariate Fractal Interpolation Functions on Grids, Fractals, 2002, vol. 10, no. 1, pp. 53-58. [9] J. E. Hutchinson, L. Rüschendorf L, Selfsimilar Fractals and Selfsimilar Random Fractals, Progress in Probability, vol. 46, 2000, pp. 109-123. [10] P. R. Massopust, Fractal Surfaces, J. Math. An. Appl, vol. 151, no. 1, 1990, pp. 275-290. [11] A. M. Navascués, Fractal Polynomial Interpolation, A. M. J. of Analysis and its Approx., vol. 24., No. 2, 2005, pp. 401-418. [12] M. A. Navascués, S. Jha, A. K. B. Chand, Generalized Bivariate Hermite Fractal Interpolation Function, Numerical Analysis and Applications, vol. 14, no. 2, 2021, pp. 103-114. [13] M. A. Navascués, J. Sangita, A. K. B. Chand, R. N. Mohapatra, Iterative Schemes Involving Several Mutual Contractions, Mathematics, 11, 2023, pp. 1-18. [14] I. Somogyi, A. Soós, Graph-directed random fractal interpolation function, Studia Mathematica, Vol 66, Nr 2, 2021, pp. 247-255. [15] H. Y. Wang, J. S. Yu, Fractal interpolation functions with variable parameters and their analytical properties, J. Approx. Theory 175, 2013, pp. 1-18.
• APA Style

Ildikó Somogyi, Anna Soós. (2023). Lagrange-type Algebraic Minimal Bivariate Fractal Interpolation Formula. Applied and Computational Mathematics, 12(5), 109-113. https://doi.org/10.11648/j.acm.20231205.11

ACS Style

Ildikó Somogyi; Anna Soós. Lagrange-type Algebraic Minimal Bivariate Fractal Interpolation Formula. Appl. Comput. Math. 2023, 12(5), 109-113. doi: 10.11648/j.acm.20231205.11

AMA Style

Ildikó Somogyi, Anna Soós. Lagrange-type Algebraic Minimal Bivariate Fractal Interpolation Formula. Appl Comput Math. 2023;12(5):109-113. doi: 10.11648/j.acm.20231205.11

• @article{10.11648/j.acm.20231205.11,
author = {Ildikó Somogyi and Anna Soós},
title = {Lagrange-type Algebraic Minimal Bivariate Fractal Interpolation Formula},
journal = {Applied and Computational Mathematics},
volume = {12},
number = {5},
pages = {109-113},
doi = {10.11648/j.acm.20231205.11},
url = {https://doi.org/10.11648/j.acm.20231205.11},
eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.acm.20231205.11},
abstract = {Fractal interpolation methods became an important method in data processing, even for functions with abrupt changes. In the last few decades it has attracted several authors because it can be applied in various fields. The advantage of these methods are that we can generalize the classical approximation methods and also we can combine these methods for example with Lagrange interpolation, Hermite interpolation or spline interpolation. The classical Lagrange interpolation problem give the construction of a suitable approximate function based on the values of the function on given points. These method was generalized for more than one variable functions. In this article we generalize the so-called algebraic maximal Lagrange interpolation formula in order to approximate functions on a rectangular domain with fractal functions. The construction of the fractal function is made with a so-called iterated function system. This method it has the advantage that all classical methods can be obtained as a particular case of a fractal function. We also use the construction for a polynomial type fractal function and we proof that the Lagrange-type algebraic minimal bivariate fractal function satisfies the required interpolation conditions. Also we give a delimitation of the error, using the result regarding the error of a polynomial fractal interpolation function.},
year = {2023}
}

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T1  - Lagrange-type Algebraic Minimal Bivariate Fractal Interpolation Formula
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DO  - 10.11648/j.acm.20231205.11
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JF  - Applied and Computational Mathematics
JO  - Applied and Computational Mathematics
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EP  - 113
PB  - Science Publishing Group
SN  - 2328-5613
UR  - https://doi.org/10.11648/j.acm.20231205.11
AB  - Fractal interpolation methods became an important method in data processing, even for functions with abrupt changes. In the last few decades it has attracted several authors because it can be applied in various fields. The advantage of these methods are that we can generalize the classical approximation methods and also we can combine these methods for example with Lagrange interpolation, Hermite interpolation or spline interpolation. The classical Lagrange interpolation problem give the construction of a suitable approximate function based on the values of the function on given points. These method was generalized for more than one variable functions. In this article we generalize the so-called algebraic maximal Lagrange interpolation formula in order to approximate functions on a rectangular domain with fractal functions. The construction of the fractal function is made with a so-called iterated function system. This method it has the advantage that all classical methods can be obtained as a particular case of a fractal function. We also use the construction for a polynomial type fractal function and we proof that the Lagrange-type algebraic minimal bivariate fractal function satisfies the required interpolation conditions. Also we give a delimitation of the error, using the result regarding the error of a polynomial fractal interpolation function.
VL  - 12
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Author Information
• Department of Mathematics and Computer Science, “Babes¸ Bolyai” University of Cluj-Napoca, Cluj-Napoca, Romania

• Department of Mathematics and Computer Science, “Babes¸ Bolyai” University of Cluj-Napoca, Cluj-Napoca, Romania

• Sections